Armature chain follow a path
Constraints


Consider a follow path constraint.  Which can rotate objects along the curve.
The curve hugger bones can have parents but must NOT be connected.  The cutting teeth can be connected to the hugger bones.  I assume the geometric normal bones are cutting teeth.
I do not think Spline Constraint is favorable.
Consider a conform modifier.  No rotation.
If you are really daring you might consider rigid body chain links.  Some people consider these situations unstable.